jankowski and squyres. sources of error in planetary photoclinometry

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 96, NO. E4, PAGES 20,907-20,922, NOVEMBER 25, 1991

Sources of Error in Planetary Photoclinometry

DAVID G. J ANKOWSKI and STEVEN W. SQUYRES

Center ]or Radiophysics and Space Research, Cornell University, Ithaca, New York

Photoclinometry has become a popular technique for the extraction of topography from digital images of planetary surfaces. The technique, however, is subject to a number of error sources that can degrade results significantly. We present here a quantitative analysis of error sources in one- dimensional planetary photoclinometry. The technique is affected by at least seven error sources, which can be broken down into three categories: those arising from (1) the spacecraft image, (2) the planetary body, and (3) the scan line orientation. Spacecraft image error sources include noise, background offset, and quantization. Errors deriving from the planetary body include those induced by variations in photometric properties, such as albedo, by incorrectly compensated atmospheric effects, and by an incorrectly determined photometric function. Finally, errors will result for scan lines which do not lie perpendicular to topographic strike. We calculate slope errors for each of these sources, using the examples of Voyager imaging of Ganymede and Viking orbiter imaging of Mars. Slope errors are investigated for a variety of viewing and lighting geometries, slope angles, and slope orientations. The results can be broken down into nonsystematic and systematic errors. Nonsystematic errors are introduced by image noise and quantization and affect the slope calculation for each picture element independently. Other error sources are systematic; these errors are more insidious, since they may retain the general appearance of the topography while approximately scaling all relief by a multiplicative constant. We present derivations that allow the calculation of photoclinometric slope errors for any photometric function and also briefly discuss the implications of our results for two-dimensional photoclinometric techniques.

INTRODUCTION

Topographic data are useful in the study of planetary surfaces and interiors. Direct measurements of topography are difficult to obtain, however, since they generally require the application of active ranging techniques. This diffi- culty has prompted the development of several techniques for the extraction of quantitative topography from spacecraft images. Perhaps the most reliable of these techniques is stereophotogrammetry [e.g., Wu, 1979; Wu and Moore, 1980]. Stereophotogrammetry can only be used on the fraction of a body's surface area for which adequate stereo cover- age is available, however, and for many solar system objects this fraction is small. The desire for quantitative topography derived from monoscopic images has motivated the development of photoclinometry.

Photoclinometric techniques make use of the surface photometric function to derive slope angles of scattered radiation from observed intensities in an image. Elevations are then calculated by "integrating" the series of slope values. The horizontal resolution of the topographic data is therefore the same as the local resolution of the image itself. The techniques most commonly in use in planetary science are one-dimensional, deriving elevations along a line of points, and the results of this paper apply primarily to these techniques. The relevance to two-dimensional photoclinometry, in which elevations are derived for a grid of points, is discussed briefly.

The earliest photoclinometric techniques were developed for use with lunar images or photoelectric scans [van Digge- len, 1951; Wilhelms, 1964; McCauley, 1965; Rindfleisch, 1966; Rowan and McCauley, 1967; Watson, 1968; Luchitta and Gainbell, 1969]. The lack of adequate radiometric cali-

Copyright 1991 by the American Geophysical Union.

Paper number 91JE02209. 0148-0227/91/91JE-02209505.00

bration of the data rendered photoclinometry unreliable for its earliest users. However, the development and use of high- quality digital imaging systems aboard planetary spacecraft has restored interest in photoclinometry, and the technique has become very popular [Mouginis. Mark and Wilson, 1981; Squyres, 1981; Howard et al., 1982; Passey and Shoemaker, 1982; Davis and Soderblom, 1984; McEwen, 1985; Moore et al., 1985; Jankowski and Squyres, 1988; Tanaka and Davis, 1988; Schenk, 1989; Plescia, 1990; Schenk, 1990]. Despite the popularity of photoclinometry, however, there has been little effort to date aimed at understanding the potential sources of error in the technique [Davis and McEwen, 1984]. Because of the growing use of planetary photoclinometry, especially for quantitative purposes, such an effort seems warranted.

One-dimensional photoclinometry is affected by seven primary error sources, which can be divided into the following three categories:

The Spacecraft Image.

Noise. Spacecraft images are not perfect representations of the radiative intensity values incident on the optical system. (We follow the convention of the astronomical commu- nity regarding radiometric terms. In this paper, intensity has units of ergs s -1 cm -2 sr -1 nm -•, and flux has units of ergs s-1 cm-2 nm- ! [e.g.,Rybicki and Lightman, 1979]. In practice, intensity in most calibrated spacecraft image data is expressed as a dimensionless ratio given by the intensity in the camera focal plane divided by the flux of the Sun divided by 7r (called I/F) [Danielson et al., 1981]. Using the National Institute of Standards and Technology radiometric definitions, intensity here is equivalent to radiance, and flux is equivalent to irradiance.) The image intensity values can include contributions from several noise sources, including detector read noise, analog-to-digital converter noise, and photon-statistic ("shot") noise. Noise causes pixel-to-pixel

20,907

20,908 JANKOWSKI AND SQUYRE$: SOURCES OF ERROR IN PLANETARY PHOTOCLINOMETRY

intensity variations that can be incorrectly attributed to topography.

Back•ound offset. The total noise behavior of an imaging system is often reasonably well approximated by Gaus- sian noise with a nonzero mean. The value of this mean

(known as the background, dark current, or pedestal), which can vary both in time through an image sequence and within a single image, must be properly subtracted if radiometri- cally accurate results are to be obtained.

Quantization. Spacecraft image data are almost always digitized, and the finite number of brightness levels results in an inherent intensity uncertainty with a standard deviation, assuming a constant probability between adjacent levels, of (2V/•) -1 data numbers (DN) (image gray level values are commonly expressed in units of

The Planetary Body

Albedo errors. Planetary surfaces nearly always contain regions with differing albedos (defined as the ratio of total power scattered by a surface element to the total power incident on the element). The inability to distinguish intensity variations due to topography from those due to variable surface albedo poses a major problem for photoclinometry. Large intensity fluctuations due to albedo variations often take shapes that cannot possibly correspond to topography. Small intensity variations, on the other hand, can sometimes be explained equally well by a gentle slope or by an albedo variation.

Atmospheric effects. Atmospheric scattering on some solar system objects can have a major effect on photoclinometric results if not accounted for properly. For images of a surface covered by a thin atmosphere, essentially all of the detected radiation travels along one of the following four paths: (1) unscattered by the atmosphere on the way down the surface and unscattered on the way up to the spacecraft; (2) scattered by the atmosphere up to the spacecraft with- out reaching the surface; (3) scattered once on the way down to the surface, and unscattered on the way up to the spacecraft; and (4) unscattered on the way down to the surface and scattered once on the way up to the spacecraft. The latter three paths introduce a constant additive term that must be subtracted (like the image background discussed earlier) from the total observed intensity.

Photometric function errors. If the surface photometric function is not known correctly, accurate photoclinometric results cannot be obtained.

The Scan Line Orientation

Scan line misalignment. In monoscopic photoclinometry, there is an inherent ambiguity caused by the fact that a single intensity value does not uniquely define a single slope orientation. For this reason, one-dimensional photoclinometric scan lines are generally restricted to lie perpendicular to topographic strike (i.e., perpendicular to the direction determined by a vector which is both tangent to the planetary surface and horizontal). As a result, one-dimensional photoclinometry is only useful for the determination of topography across regions for which the topographic strike is well determined at all points. Scan lines which are mis- aligned result in inaccurate slopes. (Note that for nonzero emission angles, choosing a scan line perpendicular to topographic strike on the surface is not the same as choosing a

scan line perpendicular to topographic strike on the spacecraft image.)

There are other sources of error that we will not consider

in detail, but users of photoclinometry should certainly be aware of them. Some of the more important ones are: viewing and lighting geometry errors, i.e., the supplementary data records (variously called SEDR's, SPICE S-, P-, and C-kernels, etc.) that accompany planetary information do not always contain accurate geometric information; oblique viewing effects, i.e., straight lines across surface relief are no longer straight when viewed obliquely (for a one-dimension al technique which accounts for this, see Davis and Soderblom [1984]); and resolution effects, i.e., topographic information at near-pixel-scale is affected by the brightness averaging in each pixel. In addition, the point spread function of the camera's optical system and detector can act to smooth out brightness variations between adjacent pixels. As a result, topography derived very near the resolution limit is not reliable.

APPROACH

Our approach to the investigation of error sources in photoclinometry is to develop an end-to-end model of the photoclinometric process. The model includes the effects of illumination and viewing geometry, the planetary scattering properties, the imaging system characteristics, and the mathematics of the topography determination process. We perform a sensitivity analysis by introducing a simulation of each of the error sources listed in the Introduction into the

model, and investigating its importance under a variety of conditions. Each error source is investigated independently, and then we perform some final calculations in which the effects of several error sources are considered simultaneously.

Five geometric parameters affect photoclinometric errors. They are the photometric latitude and longitude, the phase angle, and the surface rotation and slope angles. Photo- metric latitude and longitude are defined with respect to a photometric equator that is the intersection of the Sun- planet center-spacecraft plane with the planet's (perfectly smooth) surface. By convention, latitude 0 ø, longitude 0 ø lies at the subspacecraft point. Phase angle is defined in the usual manner, so the subsolar point lies at latitude 0 ø and longitude -c•. The surface rotation angle is the angle subtended by the photoclinometric scan line direction from the local line of photometric latitude, defined positive coun- terclockwise (Figure 1 a). The surface slope angle is defined in the vertical plane containing the scan line and is positive for surfaces tilted towards the scan direction (Figure lb).

In order to perform our calculations, we must make some specific assumptions about the scattering properties of the target body and the performance of the imaging system. We have chosen to consider Voyager imaging of Ganymede and Viking orbiter imaging of Mars. The scattering properties of Ganymede's surface are approximated using the "lunarlike" photometric function [Squyres and Veverka, 1981]

I(i.e.a) - Ff(a) [,0/(,0+,)]. (1)

where I is the scattered intensity; 7rF is the plane parallel incident solar flux; i, e, and c• are the incidence, emission, and phase angles, respectively; f(c•) is the surface phase function,/to -- cos/and p -- cos•. The scattering properties of the Martian surface are approximated by the Minnaert

JANKOWSKI AND SqUYRES: SOURCES OF ERROR IN PLANETARY PHOTOCLINOMETRY 20,909

(b)

Scanline Horizontal Direction

Tilted Surface

Figure 1.(a) Sketch of a photoclinometry scan line, showing the rotation angle •. (b) Sketch of a tilted surface along a scan line, showing the slope angle 0.

[1941] function

I(i, c, a) = F Bo po • #k-•, (2)

where B0 and k are dimensionless empirical parameters. Consider photoclinometry applied to a specific pixel on

a spacecraft image taken at a phase angle a. The location of the pixel on the (perfectly smooth) planetary surface corresponds to a particular photometric latitude •b and longitude A. If the surface at that location is horizontal, the local incidence angle i and emission angle ½ are known. If the photometric function is known, the scattered intensity I from this fiat surface is also known.

Now allow the surface to have a slope angle O. Further- more, let the pixel lie along a scan line, oriented perpendicular to topographic strike, which subtends a rotation angle ;b with respect to the local photometric latitude line. Des- ignate the local incidence and emission angles on this tilted surface as i* and ½*, respectively, and the scattered intensity as I*o

The incidence and emission angles of the tilted surface are associated with those of the flat surface via

P0 - p0 cos0 + U sin0, (3)

p* - p cos0 + V sin0, (4)

where P0 -- cos/, p -- cose, p• -- cos/*, p* -- cose*, and

U - -sin(,• + a) cos!k - cos(,• + a) sinq5 sin•b, (5)

V - -sinA cos;b - cosA simb sin;b. (6)

For the "lunarlike" photometric function, equations (3)- (6) may be solved to yield

tanO = 7- 1 v v+v , (7) ) /•o+/•

where 7 = I*/I. For error anMysis purposes, the approx- imation tan0 • 0 is adequate in equation (7). For the Minnaert photometric function, the slope angle cannot be solved for in closed form. Instead, the parameter 7 can be expanded in a power series in 0:

7 - 1 + •x0 q- •22q- f3O 3 +-". (8)

In this expansion •i -- (Oi f /OOi)s:o, where f(O) =

We can calculate the effects of various error sources in the

following manner: Designate the photoclinometrically determined slope angle as 0(x, St, z,...), where the arguments are all of the parameters that influence the c'orrectness of the result. This slope angle function can be written as a Taylor expansion about any point in parameter space:

O(2,y,z," ') 0øl(o_• c9 )" •--0

O(xo,yo, zo, " '), (9)

where 5x -- x - x0, 5y = y- Y0, and so on. The error in slope 50 due to an error in one input parameter 5x is given by

oo tiO- • •.. •,Ox" •"' (10) n=l Xo

For the lunarlike photometric function• the derivatives OnO/Ox n can be calculated directly from equation (7). For the Minfiaert function, the derivatives can be found by differentiating equation (8).

The parameter 7 can be written

I* - N- B,- A (11) Io) = T = Io--A ' where I• is the image intensity, I0 is the image intensity for attat surface, N is the image noise contribution to the measured intensity at that pixel, B is the image background at that pixel, and A is the atmospheric scattering contribution to the observed intensity value. Slope errors due to noise, background errors, atmospheric errors and albedo errors can be calculated from equations (10), (11), and either (7) or (8). Slope errors due to misalignment (rotation angle errors) can be calculated from equations (5), (6) and either

20,910 JANKOWSKI AND SQUYRES: SOURCES OF ERROR IN PLANETARY PHOTOCLINOMETRY

(7) or (8). Slope errors due to photometric function errors can be calculated from equations (8) and (10).

As an example, the photoclinometric slope error 50 caused by a background offset 5B may be calculated for the case of photoclinometry on Mars. Equations (8) and (11) can be differentiated with respect to B to give

07 OB

O0

(• + 20• + 30•3)0 B, (12)

07 7- 1 = (13)

c9B Io-B-A'

where terms up to third order have been retained in (8). Equations (12) and (13) can now be combined to give a first-order error term'

O0 7- 1 1 50(• ) -- 5B 0'-• ---- 5B I0 -- B - A •1 q' 20•2 q- 302,•3 '

(14) Differentiating (12) and (13) again with respect to B gives a second-order error term:

cgB 2 -- 5B2 •1 '{- 20•2 + 302•3

- - - + . All photometric slope errors in this paper include second- order errors. For the cases of misalignment and Minnaert k error, third-order errors are retained as well. Note that for small 0 the first-order error above can be written

5B 0 (16) 50(• ) • Io - B - A '

L•ISCUSSION

In the appendix we present detailed calculations of the magnitudes of photoclinometric errors on two bodies, Mars and Ganymede, using the approach discussed above. The major results of these calculations are summarized here. They show that the reliability of photoclinometry is dependent upon several factors: (1) The Voyager imaging system suffered from significantly lower image noise levels and background uncertainties than the Viking orbiter imaging system, resulting in lower photoclinometric slope errors. (2) Photoclinometric inversion for the Minnaert function is in-

herently less stable than for the lunarlike photometric function, resulting in unreliable photoclinometry at low incidence angles. (3) As a general rule, a larger incidence angle implies more reliable photoclinometry. Regions close to the terminator can be useless, however, due to the presence of resolved shadows and, for the case of Mars, enhanced atmospheric scattering. The optimal range is from about 15 o to 30 o from the terminator, depending upon the maximum slopes found in the region of interest. Due to foreshort- ening, large emission angles also reduce the reliability of photoclinometry. For zero emission angle, the optimal image phase angle for photoclinometry is about 600 to 75 ø. Photoclinometry at low incidence angles is completely unreliable, particularly for the Minnaert photometric function.

(4) Photoclinometric errors are strongly dependent upon the scan line rotation angle. As a general rule, scan lines should be oriented as close to parallel to local photometric latitude lines as possible. Photoclinometric results derived from scan lines at very large rotation angles tend to be unreliable, particularly for the lunarlike photometric function. (5) There are two distinct slope angle dependences present among the error sources. The noise, quantization, and albedo sources all show a very weak slope angle dependence. The background, atmosphere, misalignment, and photometric function sources, on the other hand, are all characterized by slope errors roughly proportional to the slope angle.

Systematic and Nonsystematic Slope Errors

Photoclinometric errors associated with image noise and quantization are random in nature. Although the root mean squared value of the slope errors associated with these sources can be determined, the magnitude and sign of the error for a given pixel are impossible to determine, and are independent of errors at all other points. For this reason errors associated with these two sources may be termed nonsystematic errors. For nonsystematic errors, each pixel is affected independently, resulting in the addition of a high spatial frequency component to the topographic profile.

The remaining sources may all be termed systematic errors. For systematic sources, a constant error along the scan line will produce pixel-to-pixel slope errors which add con- structively, retaining the general feature shape.

Systematic errors may affect photocl{noinetric results either by distorting isolated sections or the profile as a whole. Three of the sources, photometric function, background, and atmospheric scattering errors, generally affect scan lines as a whole. Misestimating the atmospheric scattering contribution, for example, will result in the same additive brightness error for every point of the scan line. In addition, the slope errors 50 associated with these three sources can all be approximated, for a significant fraction of the parameter space, by the relation 50 oc 0. Systematic errors of this nature dis- tort topographic profiles by approximately scaling all relief by a multiplicative constant. The qualitative appearance of the feature is retained, making detection of the distortion very difficult.

The nature of misalignment errors depends upon the nature of the topography. Misaligning a scan line across a linear feature, such as a valley, results in a constant rotation angle error along the profile; misaligning a scan line across a nonlinear feature, such as a crater, results in a variable rotation angle error. For cases in which the rotation angle error is constant along the profile, the relief-scaling effect discussed in the previous paragraph applies. For cases in which the rotation angle error is not constant, the distortion will be asymmetric along the profile.

Albedo errors, like the misalignment errors, can affect both the profile as a whole (a constant albedo offset) or isolated sections (albedo variations). The effect of an albedo error on a surface element is simpler than that of the other systematic errors. Instead of producing a slope error approximately proportional to the slope angle, albedo errors produce a slope error which is essentially independent of the slope angle. Therefore, unlike the other systematic errors, positive and negative slopes (and flat areas) are all tilted in the same direction.

JANKOWSKI AND S(•UYRES: SOURCES OF ERROR IN PLANETARY PHOTOCLINOMETRY 20,911

' I ' I

a)

actual profiles

photoclinometric profiles -

b)

c)

[ I I I I I I , 0 10 20 30 40 50

distance (kin)

Figure 2. Solid lines represent simulated photocllnometric profiles across the lunar crater Picard. Dashed profiles represent the actual topography, determined from stereophotogrammetric data [NASA, 1980, sheet 62A1S1(50)]. Vertical exaggeration is 4:1. Errors are calculated for photoclinometry on a Viking orbiter image taken at 60 o phase angle. The scan line lies at a photometric latitude of-20 ø, longitude of _$0, and has a rotation angle of 30 o . (a) Profile resulting from photoclinometry done with a 10% underestimate in the atmospheric scattering contribution, a 5-DN underestimate in the image hackground and a 0.1 overestimate in the Minnaert ]c parameter. All other errors are set to zero. (b) Profile resulting from typical Viking orbiter image noise and quantization. All other errors are set to zero. (c) Profile resulting from a 3% overestimate in the albedo everywhere along the profile. All other errors are set to zero.

Figures 2 and 3 demonstrate the effects of these different types of errors. Figure 2 shows simulated photoclinometry across an impact crater. The dashed lines give the actual topography of the lunar ciater Picard, digitized from stereophotogrammetric data [NASA, 1980, sheet 62A1S1(50)]. The vertical exaggeration is 4:1. The solid lines give synthetic photoclinometric profiles (determined by combining the stereophotogrammetrically deti•rmined slopes with slope errors calculated with our model) distorted by various error sources. The slope errors are calculated for a Viking orbiter image taken at a phase angle of 60 ø. The scan line lies at a photometric latitude of-20 ø, a photometric longitude of-5 ø, and has a rotation angle of 30 ø.

Figure 2a shows a simulated photoclinometric profile re- suiting from a 10% underestimate in the atmospheric scattering contribution, a 5-DN underestimate in the raw image background (in practice, the image background and atmospheric scattering contributions are not determined separately, their combined value may be estimated from the brightnesses of resolved shadows, which would be black in their absence) and a 0.1 overestimate in the Minnaert k pa-

rameter. All other errors are zero. The systematic nature of the three error sources is clearly evident. The crater profile is retained well, but there is about a 25% decrease in measured crater depth. If the signs of all of the error sources had been reversed, the resulting photoclinometric profile would have shown a similar increase in the crater depth.

Figure 2b shows a simulated photoclinometric profile including typical Viking orbiter image noise and quantization errors. All other errors are zero. The nonsystematic nature of the error sources is evident. The actual topographic slopes associated with crater Picard are actually quite steep compared to the random slope errors. The overall profile is not significantly altered, but spurious small-scale roughness is introduced.

Figure 2c shows a simulated photoclinometric profile re- suiting from a 3% overestimate in the albedo everywhere along the profile. All other errors are zero. As noted above, this sort of error introduces an overall slope or tilt to the profile. In practice, the albedo (more accurately, the J'(a) and B0 parameters of (1) and (2)) is a free parameter in one-dimensional photoclinometry. Its value is often determined by calculating some kind of mean over the entire scan line. This forces the profile to be horizontal "on average" disguising any regional slope.

This insensitivity to regional slopes is a clear and im-

actual profiles _ photoclinometric profiles _

0.3 - a) --

0.15 - -

0 • •

-

b)

0.3 - c) _

0.15 - -

0 • •

, I , I - I , I , 0 5 10 15 20

distance (km)

Figure 3. Solid lines represent simulated photoclinometric profiles across the lunar ridge Dorsa Ewing. Dashed profiles represent the actual topography, determined from stereophotogrammetric data [NASA, 1977, sheet 75C1S1(50)]. Vertical exaggeration is 12:1. Errors are calculated for photoclinometry under the same conditions as Figure 2. (a) Profile resulting from photoclinometry done with a 15 o misalignment error. All other errors are set to zero. (b) Profile resulting from typical Voyager image noise and quantization. All other errors are set to zero. (c) Profild resulting from a 1ø• overestimate and underestimate in the albedo of the first third of the profile. All other errors are set to zero.

:,.q 0.15

-' 0

20,912 JANKOWSKI AND SQUYRES: SOURCES OF ERROR ,N PLANETARY PHOTOCLINOMETRY

portant limitation of photoclinometry. Photoclinometry is most useful when used in constant-albedo regions containing high spatial frequencies. It is not reliable when used for low spatial frequencies (i.e., gradual regional slopes). By contrast, stereophotogrammetry is most useful when recovering low spatial frequencies. Therefore the two techniques are most effectively used to complement one another, with the photoclinometric data providing the local detail and the stereophotogrammetric data providing the regional trends.

Figure 3 shows simulated photoclinometry across a gentle ridge. The dashed lines give the actual topography of the lunar "wrinkle ridge" Dorsa Ewing, also digitized from stereophotogrammetric data [NASA, 1977, sheet 75CIS1(50)]. The vertical exaggeration is 12:1. The solid lines again give synthetic photoclinometric profiles distorted by various error sources. The slope errors are calculated for the same viewing conditions as in Figure 2.

Figure 3a shows a simulated photoclinometric profile resulting from a 150 scan line misalignment error. All other errors are zero. The systematic nature of the misalignment error is clear. The wrinkle ridge profile is retained well, but with a spurious exaggeration of the topography.

Figure 3b shows a simulated photoclinometric profile that includes Viking orbiter image noise and quantization errors. All other errors are zero. Because of the gentle topography, the nonsystematic nature of the error sources is much more evident than in Figure 2b. The topography appears to be substantially more rough than it actually is.

Figure 3c shows simulated photoclinometric profiles re- suiting from 1% overestimates and underestimates in the albedo of the first section of the profile (left of the main ridge). All other errors are zero. Although this case may seem contrived, it is not, since eolian deposition by prevail- ing winds on Mars may result in different albedos on either side of a topographic feature. For such gentle slopes, a very small albedo error can introduce a substantial and spurious topographic asymmetry.

The photoclinometric profiles in Figures 2 and 3 show the effect of a constant error source over the entire profile (or, in Figure 3c, over one section of the profile). As discussed above, some errors may vary over distances much less than the length of the profile. This is particularly dangerous for albedo variations which, in a worst case scenario, are capa- ble of distorting a flat plane into any topographic feature imaginable. In reality, albedo variations are often easy to identify, particularly when considering the entire geologic setting. It is very important, therefore, that scan lines only be chosen after a careful inspection of the region. Photo- clinometry profiles should only be produced and interpreted with full consideration of their geologic and geomorphic con- text.

Application o.f Error A naiglsis

As we have shown, there are many error sources that can affect photoclinometry, and the effects of these sources can be complicated functions of viewing and illumination geometry. Because photoclinometric results commonly are used to draw specific scientific conclusions, it is important that each photoclinometric profile have a quantitative error estimate associated with it. It is not enough to assert that the photoclinometric code in use successfully reproduces stereophotogrammetric results where stereo is available, since there is

no assurance that the same algorithm will derive accurate topography on another image under different conditions.

We have included error analysis calculations in a working photoclinometry code using the Taylor expansion technique discussed above. Due to their different natures, systematic and nonsystematic errors are considered separately.

For nonsystematic errors, a one-sigma determination of the slope uncertainty associated with image noise and digi- tation is a useful measure of the random pixel-to-pixel error:

Systematic slope errors are more complicated. As discussed above, several of the systematic sources generally affect the entire profile by approximately scaling all relief. A useful parameter for description of these errors is p = 1 + (6Osys/O), where 6Osy, is the slope error associated with the four (for linear features, three for nonlinear) relief- scaling systematic errors. The parameter p gives an estimate of the multiplicative constant that relates the derived and actual relief. Upper and lower bounds on p can be found by calculating maximum and minumum (or one-sigma) values for •Osys over the error source parameter space (q-•k, q-•B, etc.).

The freedom associated with the albedo value precludes a meaningful error estimate regarding the regional slope, and albedo variations are too case-specific to model in detail; however, albedo errors can be quantified in a general sense by calculating the slope error, •Oalo, resulting from a 5% albedo error.

The reliability of a photoclinometric profile can be quantified, then, by four parameters: •Ononsys, which gives an estimate of the size of pixel-to-pixel variations, •Oalb, which gives an estimate of the size of possible regional errors due to albedo variations, and Prna• and Prnin, which give bounds on the uncertainty of the topographic scaling of the entire profile. This analysis does not handle misalignment errors on nonlinear features nor does it account for albedo vari-

ations in detail. These are errors which cannot be easily quantified, and it is the responsibility of the photoclinometry user to minimize them. It is also, of course, the respon- sibilty of the user to make intelligent estimates of the input parameter uncertainties that determine the error estimates.

Figure 4 gives an example of this kind of error analysis applied to actual photoclinometry on Viking orbiter image 445S06. Figure 4a shows the location of a scan line chosen at a rotation angle of 80 across a • 20-km crater. In Figure 4b, the solid line is the derived topography along the scan line. The dashed lines are one-sigma error profiles determined from the Pma• and Prnin values calculated at each point. For this profile, the background uncertainty is set to -l- 5 DN, the atmospheric uncertainty is set to q- 0.002 I/F and the Minnaert k uncertainty is set to -l- 0.05. There is no assumed misalignment error (appropriate for a nonlinear feature). The one-sigma error profiles correspond to bounds on the derived profile as a unit, not as a collection of errors to independent data points. These systematic error profiles indicate the range of relief, retaining the qualitative shape, consistent with the uncertainties in the input parameters.

Figure 4c characterizes nonsystematic errors by showing 10 profiles generated by adding a component (trnons•s • 2 ø) corresponding to Viking orbiter noise and quantization to the derived profile. (By definition, all 10 profiles start at A.) The nonsystematic errors result in a pixel-to-pixel

JANKOWSKI AND SQUYRES: SOURCES OF ERROR IN PLANETARY PHOTOCLINOMETRY 20,913

A

0.5

-0.5 '• -1

0.5

o

-0.,5

-1

I

c)

-- B--

I , I , I , I , I , 0 10 20 30 40

distance (km)

Figure 4. Photoclinometry on Vikin• orbiter irna•e 445S06. (•) Image showing the location of the scan line. (•) The solid line is a photoclinolnetrically derived topographic profile. The vertical exag•/•eratlon is 10:1. The dashed lines are one-sigma error profiles, representin• the possible error in mu]tiplicative scaling of the derived relief associated with the uncertainties of the input parameters. (c) Ten profiles •enerated by adding a nonsystematic contribution to the derived profile.

vertical uncertainty of • O'nons!Is J•, where •r is in radians and/• is the image resolution. For this particular profile, the pixel-to-pixel vertical uncertainty is • 2 m, much too small to detect in Figure 4c. This is not always the case, however (e.g., Figure 3b). Although the pixel-to-pixel variations are not apparent, regional shifts between different profiles are

clearly visible in Figure 4c. On average, the 10 profiles end at the point B, as expected for nonsystematic errors. But summing up the pixel-to-pixel differences creates a regional tilt of size • O'nons•s/V/•, where N is the number of points in the scan line. Thus there are two effects assodated with

the nonsystematic errors: pixel-to-pixel vertical error bars

20,914 JANKOWSKI ANt> SQUVRES: SOURCES Or ERROR IN PLANETARY P.OTOCL•NOMZTRV

and a small regional tilt. Again, it is important to recognize that both the nonsystematic and the systematic errors are operative: so the absolute relief of the profile has an uncertainty characterized by Figure 4b, and the pixel-to-pixel slope variations introduce both a pixel-to-pixel relief uncertainty and a regional slope characterized by the spread of profiles in Figure 4c.

Two-Dimensional Photoclinometry

Applications of one-dimensional photoclinometry are limited by the requirement that the topographic strike must be known in advance. Fortunately, there are many topographic features of geologic interest for which this requirement is at least approximately satisfied. However, there are also regions where the topographic strikes are not known well enough for the use of one-dimensional techniques. Two- dimensional photoclinometry techniques, which solve for topography over an entire region rather than a line of points, circumvent this problem by attempting to solve for the topographic strike simultaneously with the slope.

Although two-dimensional photoclinometry has existed for two decades [see Horn, 1975], it has until very recently seen little use in planetary science [Kirk, 1987]. This fact is largely due to the computational complexities of the technique. Improvements in computer capabilities and advances in the technique itself will likely increase its popularity in the future, however. Horn and Brooks [1990] provide an overview of two-dimensional photoclinometric techniques.

When considering slope errors associated with two- dimensional photoclinometry, there obviously is no scan line rotation angle to misalign. However, the other six error sources are as relevant to two-dimensional photoclinometry as they are to one-dimensional photoclinometry. The one- dimensional results given here assume that each photoclinometric slope is calculated independently of its neighbors. In two-dimensional techniques, information is shared over the entire region of interest. This fact renders the quantitative results given here inappropriate for two-dimensionM photoclinometry. However, the qualitative conclusions will likely apply to both techniques. Lighting and viewing geometries that are bad for one-dimensional photoclinometry will be bad for two-dimensional photoclinometry. Slope orientations that are good for one-dimensional photoclinometry will result in accurate slopes from two-dimensional photoclinometry. Slope orientations that are bad for one-dimensional photoclinometry will produce inaccurate slopes from two- dimensional photoclinometry.

CONCLUSIONS

Photoclinometry can be a useful technique for the determination of planetary surface topography from monoscopic digital images. However, it is subject to several error sources that can affect results significantly under cer- tain circumstances. The importance of each error source is dependent upon the lighting and viewing geometries, on the orientation of the profile scan line, and on the local topographic slope angle. The reliability of the results also depends very strongly upon the ability of the person ap- plying the technique to wisely choose scan lines and accurately determine the proper photometric function, atmospheric scattering contribution, and image background contribution. For this reason photoclinometry should not be

applied as a "black box." It should only be used with a considerable familiarity with the technique, its associated errors, and the data set to which it will be applied.

Modelling of the one-dimensional photoclinometric process indicates that the reliability of the results improves with increasing incidence angle. The optimal profile location is from about 150 to 300 from the terminator. The optimal image phase angle is about 60 ø to 750 . Reliability is also very dependent upon the scan line orientation. The optimal orientation lies close to parallel to the local photometric latitude line.

Error sources may be divided into those causing nonsystematic errors, which affect each data point independently, and those causing systematic errors, which may affect the entire profile as a unit. Nonsystematic errors tend to introduce spurious pixel-scale surface roughness. Systematic errors can be particularly insidious for the unwary user, retaining the shape of a topographic profile while expanding or contracting its vertical scale.

Photoclinometry is insensitive to regional slopes. As a re- suit, the technique is most useful when applied in constant- albedo regions containing high spatial frequencies. By contrast, stereophotogrammetry is most useful when recovering low spatial frequencies. The two techniques complement each other.

Photoclinometric algorithms can readily (and, we believe, should) include error analysis calculations to help determine the reliability of the topographic profiles they generate. For every profile, it is useful to include estimates of the upper and lower bounds associated with relief-scaling systematic errors, an estimate of nonsystematic errors, and an estimate of albedo-related errors.

Topography derived using two-dimensional photoclinometry techniques are subject to most of the same errors as one- dimensional techniques, and it should be possible to subject them to similar error analyses as well.

APPENDIX

In this appendix we present detailed model calculations of photoclinometric slope errors as a function of lighting and viewing geometry. Results are displayed as contour plots in photometric latitude-longitude space. The phase angle, slope angle, and rotation angle assumed for error calculations are all held constant for a given plot. The plots extend only to 4- 600 in both latitude and longitude, since photoclinometry outside of this region is severely limited by the oblique viewing geometry.

Slope errors are first presented for each of the error sources discussed in the main body of the paper. The slope

errors are calculated for the same geometry (photoclinometry on a-10 ø slope, on a scan line with a rotation angle of 450 on an image taken at a phase angle of 60 ø) for each error source. Slope errors are then presented for one error source

(albedo variations) under varying phase, rotation, and slope angles.

In practice, the values of the dimensionless parameters

f(c•) and B0 (which are related to the local albedo) are not usually fixed in advance but instead are calculated independently for each scan line from the pixel brightnesses. For modelling purposes, we are not working with actual scan lines, and we must assume their values. For Ganymede, we


take f(a) - 0.6(1- a/lõ0)[Squyres and Veverka, 1981]. For Mars the situation is complicated by scattering in the atmosphere. We model the atmosphere as a simple homo- geneous layer in which multiple scattering is ignored. The atmosphere contributes an additive component to the image intensity (which must be subtracted off) and reduces brightness contrasts on the surface such that [Van Blerkom, 1971]: Bo - O.16exp(-r(1/po + l/p)), where r is the optical depth of the atmosphere, and the 0.16 value is determined from Viking orbiter images. The optical depth is taken to be 0.25, corresponding to typical clear Martian conditions [Pollack et al., 1977]. The phase dependence of the Min- naert k parameter for the Martian surface is taken to be k(a) = 0.45 + 0.0062a, where a is measured in degrees [Tanaka and Davis, 1988]. One-sigma error bars on k are about 4-0.05, independent of ft.

Imaging System Error Sources

The error sources deriving from imaging system characteristics are dependent upon the image exposure time, which relates the "raw" data (containing the noise, background, and quantization contributions) to the decalibrated data, such that I/F = (D- B)/?/255, where D is the raw image (8-bit) data number, B is th• background DN value, and/• is a parameter inversely proportional to the image exposure time [Klaasen et al., 1977; Danielson et al., 1981]. (Voy- ager images were 8-bit, Viking orbiter images were 7-bit. All Viking DN values in this paper have been doubled to produce equivalent 8-bit values.)

In practice, the exposure time is known for each image. For modelling purposes, we must assume a value. We make use of the fact that image exposure times are generally chosen to increase for images taken at larger incidence angles. Based on the exposure times chosen for a sample of 50 clear filter Viking orbiter images taken at incidence angles ranging from 30 ø to 72 ø, we find/•(i)•ri•: (1 + 5cosi)/5. For Ganymede, based on clear filter Voyager 1 time exposure settings, we find •(i)voy = (1 + cosi)/2. Thus the exposure time, in our model, varies smoothly across photometric latitude-longitude space. Actual exposure times assume discrete values, so a given contour plot associated with the image error sources should not be considered to represent the effects in a single image or even in a single mosaic.

Noise. The Viking orbiter and Voyager imaging systems were both dominated by read noise, so that image noise level was essentially independent of signal level. Infiight investigation of Viking orbiter images indicated that the random noise contribution had a standard deviation of 1.28 (raw image) DN, consistent with prefiight determinations [Klaasen et al., 1977]. Prefiight noise determinations for the Voy- ager vidicon cameras indicated that the random noise con-

tribution had a standard deviation of 0.75 (raw image) DN [Benesh and Jepsen, 1978].

Figures Ala and Alb show photoclinometric slope error contours due to image noise for Mars and Ganymede, respectively. The slope errors due to noise are random in nature, and the contours in Figure A1 correspond to one-sigma val-

ues. In this and subsequent figures, the dark stipple pattern marks areas beyond the terminator. The light stipple pattern marks areas for which the magnitude of the slope error excedes 10 ø .

It is clear from Figure A1 that image noise is a more significant problem for photoclinometry on Mars than it is on Ganymede. This is due to three factors: (1) the Viking imaging system was inherently noisier than the Voy- ager imaging system; (2) photoclinometric inversion using

60

'• 3O

•.. -30

-60 -60 -30 0 30

Photometric Longitude (deg)

Fig. Ala

6O

60

3O

-30

I I I

•o

! .5 ø

Ganymede

-60 -60 -30 0 30 60


Fig. Alb

Figure A1. Slope error contours associated with photoclinometry due solely to image noise. Errors are calculated for a-10 ø slope on a scan line having a rotation angle of 450 on mn image at taken at 600 phase angle. The dark stipple pattern designates area beyond the terminator. The liõht stipple pattern designates areas with slope errors exceding 10 . (a) Contours for Mars due to typical Viking orbiter image noise. (b) Contours for Ganymede due to typical Voyager image noise.


the Minnaert function is less stable than that using the lunarlike photometric function; and (3) atmospheric scatter- inõ reduces the brightness contrasts attributable to Martian surface topography. Figure Ala indicates that noisy photoclinometry on Mars is more reliable at positive than negative photometric latitudes. This asymmetry is a result of the assumed rotation and slope angles. For this rotation angle, scan lines at positive latitudes are oriented closer to perpendicular to local isophotes than scan lines at negative latitudes. Scan lines oriented along isophotes tend to have enormous errors. Conversely, noisy photoclinometry

6o

Quantization

3O

-30

-60 -60 -30 0 30 60


Fig A2a

6O

• 0

.

a.. -30 0.5"

• 0.25* -60 I • •

-60 -3O 0 3O 6O Photometric Longitude (deg)

Fig A2 b

Figure A2. Slope error contours associated with photoclinometry due solely to image brightness quantization. Errors are calculated for a scan line under the same conditions as in Figure A1. (a) Contours for Mars due to Viking orbiter image brightness quantization. (b) Contours for Ganymede due to Voyager image brightness quantization.

6O

,• 30

.•..•

3 o .,.•

o

o

•., -30

-6O -60

--T I 1 I

-30 0 30 Photometric Longitude (deg)

Fig. A3a

Background

6O

3O

•.. -30

Ganymede

Background

0.5 ø

0.4 ø

).2'

0.3 ø

6O

Figure A3. Slope error contours associated with photoclinometry due solely to incorrect image background subtraction. Errors are calculated for a scan line under the same conditions as in Figure A1. (a) Contours for Mars due to a 4-DN underestimate in the raw image background. (b) Contours for Ganymede due to a 2-DN underestimate in the raw image background.

on Ganymede is more reliable at negative photometric latitudes due to the very different natures of the two photometric functions. At a given latitude for either case, slope errors tend to decrease at larger longitudes. This is not true for regions on Mars close to the terminator, however, where atmospheric scattering begins to dominate surface scattering.

Quantization. Figures A2a and A2b show photoclinometric slope error contours due to quantization. As in Fig- ure A1, the contours in Figure A2 correspond to one-sigma

-60 -60 -30 0 30 60


Fig. A3b

JANKOWSKI AND SQUYRES: SOURCES OF ERROR ,N PLANETARY PHOTOCLINOMETRY 20,917

vMues. Since the effect of quantization is equivalent to that of noise, Figures A1 and A2 are qualitatively very similar.

Back9round offset. The uncertainty in the Viking background is about 4-5 (raw image) DN (E. Eliason, personal communication, 1990). (In practice, this can be lowered significantly in many cases by finding images with shadows.) The uncertainty in the Voyager dark current is much smaller, typically about 0-2 (raw image) DN, depending upon the availability of dark images (W. R. Thompson, personal communication, 1990).

6O

3O

-30

4 ø

-60 -60 -30 0 30 60


Fig. A4a

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-30

3 ø

-60 -60 -30 0 30 60


Fig. A4b

Figure A4. Slope error contours associated with photoclinometry due solely to an incorrect surface albedo. Errors are calculated for a scan line under the same conditions as in Figure A1. (a) Con- tours for Mars due to a 5% overestimate in albedo. (b) Con[ours for Ganymede due to a 5% overestimate in albedo.

Figures A3a and A3b show photoclinometric slope error contours due to an underestimate in the background of 4 DN for Mars and 2 DN for Ganymede, respectively. A background offset clearly affects photoclinometry differently than do noise and quantization. For background offset, slope errors tend to increase toward the terminator for both cases.

Figure A3a indicates that photoclinometric inversion with the Minnaert function becomes unstable for a narrow strip of area at large negative latitudes and longitudes.

Planetary Body Error Sources

Albedo errors. Figures A4a and A4b show photoclinometric slope error contours due to an overestimate in the

albedo of 5%. For negative photometric longitudes, the error contours in Figure A4a are similar to those in Figure Ala. For positive longitudes, the error contours differ significantly. The slope errors continue to decrease towards the

terminator in Figure A4a since the atmospheric scattering reduces the error source along with the topographic brightness contrasts.

Atmospheric effects. Figure A5 shows photoclinomet- tic slope error contours due to a 20% overestimate in the atmospheric scattering contribution for Mars. Errors are calculated for the same phase, rotation, and slope angles as in Figure A1. Figures A5 and A3a resemble one another closely due to the additive nature of their error sources.

Photometric ]unction error. Slope errors associated with a photometric function error are calculated for Mars

under the assumption that the Minnaert k parameter (and the corresponding B0) is incorrect everywhere along the profile. Figure A6 shows photoclinometric slope error contours due to an overestimate of 0.1 in the Minnaert k parameter for Mars. Figure A6 demonstrates the very different nature of this error source compared to those discussed previously.

6O

'• 30

o o

o

o

• -30

-6O -60

_4 ø

A t toosphere

-3O 0 30 60 Photometric Longitude (deg)

Figure A5. Slope error contours associated with photoclinometry on Mars due solely to a 20% overestimate in the atmospheric surface contribution. Errors are calculated for a scan line under

the same conditions as in Figure A1.


60

30

-30

-60 -60

I I

-30 0 30 60 Photometric Longitude (deg)

Figure A6. Slope error contours associated with photoclinometry on Mars due solely to a 0.1 overestimate in the Minnaert k parameter. Errors are calculated for a scan line under the same conditions as in Figure A1.

The same •k can result in photoclinometric slope errors that are either positive or negative, depending on the photometric latitude and longitude.

Scan Line Orientation Error Source

Figures A7a and 7b show photoclinometric slope error contours due to a 100 misalignment error. The slope error due to scan line misalignment is affected by the orientation of the scan line with respect to the local isophote. As the scan line orientation approaches the isophote orientation, the misalignment slope error increases. Figure A7a demonstrates that slope errors are lower at positive latitudes than at negative latitudes, consistent with all other error sources for Mars. Figure A7b shows the simple nature of the lunarlike photometric function, which results in isophotes with a shape independent of photometric longitude. The photoclinometric slope errors are lower at negative latitudes than at positive latitudes, consistent with all other error sources

for Ganymede for the assumed rotation and slope angles. The following plots investigate the effects of the phase an-

gle, the rotation angle, and the slope angle. All calculations are done for photoclinometry with a 5% albedo overestimate.

Phase Angle Dependence

The slope error calculations shown in Figures A1-A7 assume an image phase angle of 60 ø. Figures ASa and ASb show contours for photoclinometry on Mars at phase angles of 300 and 90 ø, respectively. Figures ASc and ASd show contours for photoclinometry on Ganymede at phase angles of 300 and 900 , respectively.

A comparison of Figures ASa, ASb, and A4a shows that photoclinometric slope errors on Mars due to albedo variations at a given distance from the terminator tend to de-

crease slightly with increasing phase angle. A comparison of Figures ASc, ASd, and A4b shows that the same result holds for photoclinometric slope errors due to albedo variations on Ganymede. The slope error change is small in both cases, however; the slope error at a given distance from the terminator is not strongly dependent upon the image phase angle. Although the errors in Figure A8 are calculated for an albedo error, the same qualitative result holds for the other error sources as well.

6O

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-30

-60 -60

1 I I I i

_1 ø

-2'

-3*

-30 0 30


Fig. A7a

Misalignment

60

6O

'• 3O

-30

-2*

-60 -3O 0 30 Photometric Longitude (deg)

Fig. A7b

60

Figure A7. Slope error contours associated with photochnometry due solely to an incorrect rotation angle. Errors are calculated for a scan hne under the same conditions as in Figure A1. (a) Contours for Mars due to a 100 misalignment error. (b) Contours for Ganymede due to a 10 ø misalignment error.


30 '• 30

0 • 0

-30 a.., -30

! o •. _... o

-6,0 -60 -6,0 -30 0 30 6,0 -60


Fig. ASa

-30 0 30

Photometric Longitude (deg) Fig. ASh

6O

30

0

-30

-60 -60 -30 o 30 60


60 •

3O

-30

-6O -60

3 ø


Fig. A8c Fig. A8d

Figure A8. Slope error contours associated with photoclinometry due solely to an incorrect surface albedo. Image phase angles differ from that of Figures A1-A7, but all other conditions remain the same. (a) Contours for Mars at a phase angle of 30 ø . (b) Contours for Mars at a phase angle of 90 ø. (c) Contours for Ganymede at a phase angle of 30 ø . (d) Contours for Ganymede at a phase angle of 90 ø.

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Rotation Angle Dependence

The slope error calculations shown in Figures A1-A7 assume a rotation angle of 45 ø. Figure A9a and A9b show contours for photoclinometry on Mars at rotation angles of 0 ø and 90 ø, respectively. Figure A9c and A9d show contours for photoclinometry on Ganymede at rotation angles of 0 ø and 80 ø, respectively. Note that the error contours in Figures A9a and A9c are symmetric about the photometric equator, as expected for a rotation angle of 0 ø.

A comparison of Figures A9a, A9b, and A4a shows that

photoclinometric slope errors due to albedo variations on Mars are strongly dependent upon the rotation angle. This is particularly true at large rotation angles, where slope errors become large almost everywhere. A comparison of Fig- ures A9c, A9d, and A4b shows that this same result holds for photoclinometric slope errors due to albedo variations on Ganymede. In fact, if Figure A9d were plotted at a rotation angle of 900 instead of 800 , the slope error would excede 10 ø over the entire plot. Although the errors in Figure A9 are calculated for an albedo error, the same qualitative result again holds for the other error sources as well.

20,920 JANKOWSKI AND SQUYRES: SOURCES Or ERROR IN PLANETARY PHOTOCLINOMETRY

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- 6*

4 ø

-30 0 30 Photometric Longitude (deE)

Fig. A9a

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4 ø

C ø

_6 ø

_4 ø

_2 ø

-30 0 30 60 Photometric Longitude (deE)

Fig. A9b

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-30

-60 -60

1 ø

rotation--O ø

-30 0 30 60 Photometric Longitude (deE)

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-60 -60

......................................... ••••••••••:.._..•.

.•...- -. ß -...._..._..._.._.......•..• .•-••0••••.•••.•••••••:.•

.......................................... 1 o

I

-30 0 30 60 Photometric Longitude (deg)

Fig. A9c Fig. A9d

Figure A9. Slope error contours associated with photoclinometry due solely to an incorrect surface albedo. scan line rotation angles differ from that of Figures A1-A?, but all other conditions remain the same. (a) Contours for Mars at a rotation angle of 0 ø. (b) Contours for Mars at a rotation angle of 90 ø. (c) Contours for Ganymede at a rotation angle of 0 ø. (d) Contours for Ganymede at a rotation angle of 80 ø.

Slope Angle Dependence

The slope error calculations shown in Figures A1-A7 assume a slope angle of-10 ø. Figures A10a and A10b show contours for photoclinometry on Mars at slope angles of 0 ø and 10 ø, respectively. Figures A10c and A10d show contours for photoclinometry on Ganymede at slope angles of 0 ø and 10 ø.

A comparison of Figures A10a, A10b, and A4a shows that photoclinometric slope errors due to albedo variations on Mars are not strongly dependent upon the slope angle. At a given distance from the terminator, slopes facing the

Sun have slightly lower incidence angles than slopes facing the terminator, resulting in higher slope errors for the negative (sunward facing) slopes. A comparison of Figures A10c, A10d, and A4b shows that the same result holds for photoclinometric slope errors due to albedo variations on Ganymede.

Unlike the phase and rotation angle dependences, the slope angle dependence varies significantly among the various error sources. The weak slope angle dependence found in Figure A10 for the albedo variation also holds for the noise and quantization sources. The remaining four sources, background offset, atmospheric effects, photometric error,

JANKOWSKI AND SQUYRES: SOURCES OF ERROR 'N PLANETARY PHOTOCLINOMETRY 20,921

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•., -30

-6O

2 ø

6O

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4 ø

-60 -30 0 30 60 -60 -30 0 30 Photometric Longitude (deg) Photometric Longitude (deg)

Fig. A10a Fig. A10b

-6O -30 0 30 60 Photometric Longitude (deg)

60

30

-30

-6O -60

--T--T-- I 1 [


Fig. A10c Fig. A10d

Figure A10. Slope error contours associated with photoclinometry due solely to an incorrect surface albedo. Slope angles differ from that of Figures A1-A7, but all other conditions remain the same. (a) Contours for Mars at a slope angle of 0 ø. (b) Contours for Mars at a slope angle of 100 . (c) Contours for Ganymede at a slope angle of 0 ø . (d) Contours for Ganymede at a slope angle of 10 ø.

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and misMignment, have a similar slope angle dependence. For these four sources, the slope error is roughly propor- tionM to the slope angle for small slopes.

Acknowledgments. We are grateful to Eric Eliason, Paul Heffenstein, and Reid Thompson for helpful discussions and to Paul Schenk and an anonymous referee for very insightful reviews. This work was supported by the NASA Planetary and Geology and Geophysics Program.

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(Received November 13, 1990; revised August 7, 1991;

accepted August 26, 1991.)

jankowski and squyres. sources of error in planetary photoclinometry

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